Efficient Variable-Coefficient Finite-Volume Stokes Solvers
Efficient Variable-Coefficient Finite-Volume Stokes Solvers
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DOI：
10.4208/cicp.070114.170614a
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发表时间：
2014-11-01
影响因子：
3.7
通讯作者：
Donev, Aleksandar
中科院分区：
文献类型：
--
作者：
Cai, Mingchao;Nonaka, Andy;Donev, Aleksandar
研究了在均匀交错网格上由非定常和定常变系数Stokes方程空间离散引起的鞍点线性系统的几个鲁棒预条件。在经典投影法作为速度-压力耦合系统预调节器成功的基础上[B]。E.格里菲斯，J.康普。[j]，第228(2009)，第7565-7595页]，以及在有限元文献中建立的稳定和非稳定Stokes流技术，我们构建了使用独立的广义Helmholtz和泊松解速度和压力子问题的预条件。我们证明了只有一个标准几何多网格算法的一个循环可以作为这些子问题的有效非精确解算器。与传统智慧相反，我们发现Stokes问题几乎可以像独立的压力和速度子问题一样有效地解决，使得解决Stokes系统的总体成本与经典投影或分步方法的成本相当不可压缩流动，即使是在稳定流动和存在大密度和粘度对比的情况下。本文考虑的五个预调节器中有两个对GMRES重启和增加问题规模具有鲁棒性，使其适用于大规模问题。我们的工作为构建低马赫数和不可压缩流动动力学方程的有限体积空间离散化的新型非分裂时间积分器提供了许多可能性。
We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.